Performance issue with generation of random unique numbers

Question

I have a situation where by I need to create tens of thousands of unique numbers. However these numbers must be 9 digits and cannot contain any 0\'s. My current approach is

Answer 1

Something like this?

public List generateIdentifiers2(int quantity)
        {
            var uniqueIdentifiers = new List(quantity);
            while (uniqueIdentifiers.Count < quantity)
            {
                var sb = new StringBuilder();
                sb.Append(random.Next(11, 100));
                sb.Append(" ");
                sb.Append(random.Next(11, 100));
                sb.Append(" ");
                sb.Append(random.Next(11, 100));

                var id = sb.ToString();
                id = new string(id.ToList().ConvertAll(x => x == '0' ? char.Parse(random.Next(1, 10).ToString()) : x).ToArray());

                if (!uniqueIdentifiers.Contains(id))
                {
                    uniqueIdentifiers.Add(id);
                }
            }
            return uniqueIdentifiers;
        }